Question

The following is an incomplete flowchart proving that the opposite angles of parallelogram JKLM are congruent:

Parallelogram JKLM is shown where segment JM is parallel to segment KL and segment JK is parallel to segment ML.

Extend segment JM beyond point M and draw point P, by Construction. An arrow is drawn from this statement to angle MLK is congruent to angle PML, Alternate Interior Angles Theorem. An arrow is drawn from this statement to angle PML is congruent to angle KJM, Corresponding Angles Theorem. An arrow is drawn from this statement to angle MLK is congruent to angle KJM, Transitive Property of Equality. Extend segment JK beyond point J and draw point Q, by Construction. An arrow is drawn from this statement to numbered blank 1, numbered blank 2. An arrow is drawn from this statement to angle QJM is congruent to angle LKJ, Corresponding Angles Theorem. An arrow is drawn from this statement to angle JML is congruent to angle LKJ, Transitive Property of Equality. Two arrows are drawn from this previous statement and the statement angle MLK is congruent to angle KJM, Transitive Property of Equality to opposite angles of parallelogram JKLM are congruent.

Which statement and reason can be used to fill in the numbered blank spaces?


∠QJM ≅ ∠JKL
Alternate Exterior Angles Theorem

∠JML ≅ ∠QJM
Alternate Exterior Angles Theorem

∠QJM ≅ ∠JKL
Alternate Interior Angles Theorem

∠JML ≅ ∠QJM
Alternate Interior Angles Theorem

Answer 1

∠JML ≅ ∠QJM

Alternate Interior Angles Theorem

Explanation:

Answer 2

Answer:∠JML ≅ ∠QJM

Alternate Interior Angles Theorem

Explanation:

Here, Given: JKML is a parallelogram,

That is, JK ║ ML and JM ║ KL

Prove: ∠ MLK ≅ ∠ KJM and ∠ JML ≅ ∠ LKJ

Extend segment JM beyond point and draw point P (Construction)

∠ MLK ≅ ∠ PML ( Alternate Interior Angles Theorem)

∠ PML ≅ ∠KJM ( Corresponding Angles Theorem )

⇒ ∠ MLK ≅ ∠ KJM ( By transitive property of equality )

Extend segment JK beyond point J and draw point Q, by Construction.

∠JML ≅ ∠QJM ( Alternate Interior Angles Theorem)

∠ QJM ≅ ∠ LKJ ( Corresponding Angles Theorem )

⇒ ∠JML ≅ ∠ LKJ ( By transitive property of equality )

Thus, the opposite angles of parallelogram JKLM are congruent.